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The Melencolia Code David R. Finkelstein School of Physics Georgia Institute of Technology March 23, 2005 1

Melencolia Code by Finkelstein

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The Melencolia Code

David R. FinkelsteinSchool of Physics

Georgia Institute of Technology

March 23, 2005

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1 MELENCOLIA I

The imposing angel stares intently with a slight smile at nothing in the picture. An oddly-shapedpolyhedron balances her on the left. A lizard-tailed bat-winged creature bears the cryptic misspeltmotto ”MELENCOLIA I.” A moon-bow frames the night sky over a dark sea. A winged spirit, orputto, writes on a block, perched on a rug draped over an up-turned stone wheel. Devices of thecrafts and sciences clutter the scene, including a magic square. A gaunt dog curls up between thepolyhedron and a globe.

The Melencolia I (1514) of Albrecht Durer (1471-1528) might be the most studied engraving evermade, and it influenced European art for centuries [12, 19]. It represents science well before Newton.Many riddlers have tried to decode it. The art historian Erwin Panofsky [12] saw in it Durer’s ownmelancholy frustration at the gap between artistic and divine creation. Frances Yates, historianof the Hermetic tradition, took its melancholia to be an inspired creative fever, not sadness at all,and read the engraving as a declaration of the harmony between microcosm and macrocosm [18].The art historian Patrick Doorly sees it as an illustration for Plato’s Greater Hippias, a dialogue onbeauty [2]; the angelic melancholy represents the inability to define absolute beauty. Long beforeI heard of their studies, I saw in it a feeling about science that I could not quite read. Was theangel truly melancholy? If so, was it for knowing too little? Or too much? Is the angel dreamingof a Final Theory? Isn’t she actually smiling slightly? What is the joke?

Panofsky, Yates, and Doorly concern themselves with what Durer says about art than science, butDurer did not distinguish art and science as sharply as we do today, and in any case he was aneminent mathematician as well as an artist. What I propose here confirms some of their overlappingunderstandings. But I also find a host of concealed messages in the engraving that they do notmention.

There is no doubt that Durer literally encoded messages into work of this period. His TriumphalArch for Maximilian I was done in the same year and it is rampant with allegorical symbolism. Itincludes an elaborate eulogy of Maximilian that was encoded in stages. First Willibald Pirckheimer(1470-1530), Durer’s learned best friend, Humanist, councillor, and translator of Greek and Hebrewclassics, wrote it in Latin. Then Durer translated it literally into a language composed of allegoricalimages of dogs, goats, harpies, cocks, snakes, scepters and the like. Durer took most of his sym-bols from the Hieroglyphica, a completely spurious but widely accepted decoding of the Egyptianhieiroglyphics. This was allegedly written in Greek by one Horus Apollo of the second or fourthcentury. Pirckheimer translated the Hieroglyphica into Latin and Durer illustrated it in 1512/1513[12]. The key to the Triumphal Arch code was thus recorded and passed down to us today; butnot the key to the Melencolia I code. Besides the kind of symbolism permeating the TriumphalArch, Melencolia I has hidden faces, anagrams, rebuses, Cabalistic gematria, puns, metaphors, anda magic charm, not mentioned by Panofsky and Yates. We have to discover and decipher these forourselves.

There are well-known limits to Durer’s knowledge. Most famously, as the astronomer Kepler pointedout, in Durer’s work on measurement and human proportions, he draws the curve produced bycutting a cone with a plane as egg-shaped, with its small end near the tip of the cone. Dureralso commited a subtle error in atmospheric optics in this engraving. Nevertheless, to catch his

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meanings we should respect his power with the burin, his concern for precise language, and hisacute observations of nature. We can then read the latent images in Melencolia I rather clearly.

Decoding his message requires some background and takes us back to many beginnings. The firstpublished art criticism in history was written in praise of just this engraving by the first publishedart critic, Joachim Camerarius (1500-1574). Camerarius, a friend of Durer, watched him at work,translated his book on human proportions from German into Latin, and wrote his eulogy. In it hedeclares that although Durer’s art was great, it was the least of his accomplishments. To read Durerwith the respect he deserves, we must remember that he was a pioneer in mathematics as well asart, a poet and an antiquarian, and argued philosophy hotly and on equal terms with scholars likePirckheimer.

The work also has mementos of Leonardo da Vinci (1452-1519). Leonardo was one of the early ex-ponents and masters of the perspective technique and summarized the basic ideas in his Notebooks.Durer studied “the secret of perspective” and the theory of human proportion on two historic visitsto Italy, developed a passion for geometry, formulated mathematical concepts of beauty and pro-portion extending Leonardo’s work, and is credited by some with the new mathematical science ofprojective or descriptive geometry.

Descriptive geometry was the first mathematical theory of relativity, in the general sense of a theoryof how different people perceive the same objects or processes. Like special relativity, perspectivetheory is founded on the propagation of light, so there are several family resemblances betweenthe two theories. Where relativistic physicists today speak of “light cones,” Leonardo and Durerspoke of “pyramids of vision,” one with apex at the the observing eye like a past light cone, andone at the point observed like a future light cone. Their pyramids of vision are our light-coneswith time left out. The perspective transformations of the Renaissance are closely related to therelativity transformations of modern physics. The famous foreshortening of limbs in perspectivehas the same nature as the equally famous shortening of rulers and slowing of clocks in relativemotion. Perspective theory and relativity theory use transformations of the same family (the lineargroup in two dimensions, real or complex respectively).

Durer was the first to publish in German a book on pure mathematics or a mathematical proof.He produced the first printed star-chart and the first printed map of the world as a sphere. Bydescribing the processes of art and nature in German instead of Latin, and dropping the proseornamentations of the time, he formed the German scientific language much as Leonardo hadformed the Italian.

He was a brilliant and acknowledged originator in art too. He invented etching. He did the firstknown self-portrait, and the first known specific landscape. He invented a mechanical frame forperspective drawing.

His life’s mission was to sanctify the artist above the artisan and even the philosopher. Since itseemed to be their mathematical structures that made astronomy and music divine and part ofthe quadrivium, Durer tried to construct mathematical foundations for art. Ptolemy had providedmathematical foundations for astronomy, and Archimedes for music; Durer wished to do so for art.Projective geometry and the measurement of body proportions are both contributions by him tothis end.

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He seems to have been the first artist to conceive of his special talent as a divine gift and chargefor it accordingly, instead of just billing for time and materials like Raphael or any housepainter ofthe time.

I have divided the decoding of Melencolia i into twenty elements that any interpretation shouldexplain.

2 THE MYSTERY

Why all the mystery? Durer could have explained his engraving for us. Leonardo was a greatriddler, but he wrote the solutions next to his riddles. Durer deciphered his Triumphal Arch forus; why not this engraving? Certainly much of the picture’s hold on us depends on its mystery,but some of the concealments are too thorough to have artistic purpose. In his essay on this work,Camerarius too perpetuated its opaqueness when he described the magic square full of numbers asa spiderweb full of dead flies, ignored the many hidden elements, and omitted the religion and thescience [8].

I suppose that they were less than frank because when they worked the thought police were per-secuting both the old magic and the new sciences with increasing ferocity. Accusations of blackmagic and witchcraft beset magicians and natural philosophers alike. The astrologist Heinrich Cor-nelius Agrippa von Nettesheim, who may be represented in the engraving, was eventually accusedof witch-craft by Luther. In 1530, Calvin would (mistakenly) accuse Michael Servetus, discovererof the cardio-pulmonary circulatory system, of writing that travellers found Palestine barren com-pared to the biblical account, and burn him and his book at the stake in Geneva. The astronomerJohannes Kepler and his mother were tormented as late as 1615 by accusers who denounced Ke-pler’s science-fiction story of a trip to the Moon as evidence of witchery. When we have read Durer’smessage, we will understand his reticence and that of his heirs.

3 THE GHOSTS

What ghosts?

What triggered my own new interest in Melencolia I and Durer were two faces that I found hidingin the polyhedron in April 2004. Anyone who steps back several paces from a good print andfocuses on the shading of the front face of the polyhedron patiently for a minute or so, will soonfind or construct a face, either a man with his head cocked to the right or a woman with her headerect. Both are there, at different angles and scales, apparently waiting to be seen for centuries.Some viewers see the man first and some the woman; one cannot see both at once. Both disappearif we come too close. Digitizing and reducing the engraving for the internet sometimes outlines oneof the faces crisply, bringing it completely out of hiding. That is how I found the woman by chanceand was drawn into this study. Absent such electronic aids, the faces emerge from the polyhedronslowly the first time but are inescapable thereafter. They are subtle and draw much on our ownperceptive process, so that as we see them taking shape, we are not entirely clear whether they are

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really there or are our own projections. One can resolve these doubts to some extent. We may notethat we cannot find such faces in all of Durer’s shadings. We can verify the faces we see with otherviewers. We may assume that Durer’s vision was more sensitive to variations of shade than most,and infer that if we see these faces then so did Durer, and therefore they are part of his intention.

In 1604, long after popular demand had worn out Durer’s original copper plate, the Dutch engraverJan Wierix (ca. 1553-1619) produced a new engraving of Melencolia I from scratch, so to speak.In order to distinguish his copy from the original he left out a flourish between “Melencolia” and“I” in the motto. Less publicly, he also systematically changed both hidden faces to hidden devil’smasks. By this change he left a secret sign that he saw the hidden faces and rejected what theystood for. This convinces me that they are not of my creation. And later they will lead us to thefunction of the polyhedron.

Durer’s other works also contain many hidden faces in many techniques. His watercolor View ofArco has a cliff that clearly is also a comically scowling face. His engraving variously known as DerSpaziergang , The Lady and the Gentleman, and Young Couple Threatened by Death, reproducedon page 201 of Panofsky [12], has the most astonishing hidden face of them all, formed of andlooming over the lady and the gentleman in question. In some ways Pirckheimer came betweenAlbrecht and his wife Agnes. For example, Albrecht left Agnes in Nuremberg immediately aftertheir honeymoon and went to Italy, where Pirckheimer was at the time. I suggest uncertainly thatthe lady and the gentleman might be Agnes and Albrecht, the hidden face Pirckheimer, and thefigure of Death behind the tree, the plague. We will also find a significant hidden face in Durer’sengraving The Knight, Death, and the Devil . Sometimes the hidden face furthers the story of thepicture like a Greek chorus. See the astonished face high in the tree-top over Durer’s Orpheus theFirst Pederast (or, The Death of Orpheus) watching a woman beat Orpheus to death with a branch.

One can read the surface of the polyhedron in three ways: as blank, as a woman’s face, or as aman’s face. This is the only such triple image I find in Durer’s work. The two hidden faces resemblesomewhat Durer’s last portraits of his mother and father, and have the same poses. Durer’s fatherdied in 1507 at the age of 75 and a Durer portrait shows him beardless at 70, but five years isenough for the small beard on the hidden face of the man. Durer did his last portrait of his mothertwo months before her death and Melencolia I soon after. There is not enough detail in the hiddenfaces for positive recognition but since the ages, poses, and general features match, and the picturehas already been recognized as autobiographical, perhaps they represent his mother and father.They have another function that I will point out later. They are part of a joke that Durer isplaying on us.

I am less certain of a third hidden face. The black axle-hole in its center of the millstone might passfor a silhouette profile of a lean young man, presumably Albrecht again. Unlike the two parentalghosts, this hidden face reverses figure and ground, a common way to hide faces at the time.

4 THE REBUS

There is one rebus in the engraving, and two in the Durer coat-of-arms. What do they mean?

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The famous Durer monogram formed from the initials AD appears under the date 1514 on the edgeof the stone slab under the angel. It seems to pun on Anno Domini, as if to equate Durer andDomini and to subordinate his family name to his personal one. But the capital A is not just apersonal initial but is also a cryptic sign. To decipher it we need some family background.

In 1455 one Albrecht of Ajtas, Hungary, goldsmith and artist, moved to Nuremberg, Germany.Ajto is Hungarian for door, or Thur in the German of the time, so Albrecht of Ajtas becameAlbrecht Thurer, which may be rendered as “Albert Man-of-the-Door.” Had he moved to Englandhe might have become Albert Gates. He soon simplified the spelling to Durer, which sounded thesame in one dialect. He gave his name to his son, born in Nuremberg, who became the famousartist-mathematician.

Albrecht the Younger drew up a family coat of arms that tells us something about his self-conceptionand helps us to read Melencolia I . Its central element is a gateway with two open doors, clearlya rebus for the family name and origins. This leads us to look for another rebus. One is closeat hand. The coat of arms includes conventional symbols that can be found in any dictionary ofheraldry. The closed helmet in full profile stands for the rank of esquire, the lowest; the eagle wings,for fame and glory; and the blackamoor, for heroic action in the Crusades. Then there is a crypticbut significant ornament at the very bottom of the coat of arms. The whole gate stands on a cloud.This is rather improbable, and so carries much information. I read it as follows.

Coats of arms usually show off the pride of the family. The cloud too could be a rebus for somegreat personal accomplishment. To proclaim what he was most famous for, Durer could simply haveadded the engraver’s tool, a burin, to his coat of arms. But the burin is also a mere artisan’s tool,and Durer’s life mission was to sanctify the artist above the artisan. Albrecht presumably knewthe Latin for his burin, which is caelum; Pirckheimer was a scholar of classic languages. Caelumalso means “the sky, ” “heaven,” “the heavens.” It has the same root as our “celestial.”

Durer did not miss this opportunity for self-sanctification. It seems virtually certain that he puthis gates “in heaven” (caelo) , as a punning rebus for “I engrave” (caelo). This is our second rebus.We will see further evidence for this reading in the engraving Melencolia I itself.

An educated person of the time might have taken this claim of divinity as a pious affirmation thatwe are made in the image of God, and not necessarily as a sacrilegious display of ego. The coatof arms can be read as “Porta caelo,” gate of heaven. Since the doors are open, we mainly see agateway in the sky, or limen caelo; I will explain later why I prefer the latter reading.

With Durer’s coat of arms in mind, one now reads the A in his monogram as a gateway as well asa letter. The legs of the gateway are the uprights of the A, the lintel connects them at the top,and the stiffener just beneath the lintel is the cross-bar of the A . The polymorphic A-gate is ourthird rebus for the Durer family. This A is the first of several gates we will find in the picture itself,important for its interpretation.

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5 THE MAGIC SQUARE

What does the magic square mean? The matrix of the numbers from 1 to 16 set into the wallbeyond the angel is called a magic square because every row, every column, and both diagonalsadd up to the same number, 34. It is also a gnomon magic square, because in addition so do all fourquadrants; the four corner numbers; and the central tetrad. The idea of the magic square couldhave come to Durer from Heinrich Cornelius Agrippa von Nettesheim (1486-1535), a controversialfigure in the Florentine Neo-Platonist school who visited Nuremberg in 1510 with the manuscriptof his De Occulta Philosophia. In that work Agrippa gave a magic square for each of the seven“planets” then known, including the Sun and the Moon. He reserved the 1-by-1 magic square 1for the one God. He kept the “planets” in their Babylonian order, which is nearly the order of theirperiods. There is no 2-by-2 magic square, so Saturn, the slowest planet, had a 3-by-3 table. Thenext slowest planet is Jupiter, so Jupiter’s Table was a 4-by-4 table, namely

4 14 15 1

9 7 6 12

5 11 10 8

16 2 3 13

This magic square, often written with Hebrew numerals instead of Arabic, is still used as a talismanin modern times. Agrippa warned that the undiluted influence of Saturn could lead to acutemelancholia and prescribed wearing Jupiter’s Table as shield. He argued his case with gematria, aCabalistic practice based on the fact that any letter of the Hebrew alphabet is also a number.

The magic square in Melencolia I is

16 3 2 13

5 10 11 12

9 6 7 12

4 15 14 1

Panofsky and Yates call this Jupiter’s Table, suggesting that Durer borrowed it form Agrippa. Ifwe look at the numbers in the table we see that it is clearly not Jupiter’s Table, so let us call itDurer’s Table. By a succession of elementary operations on the Table, Durer changed the bottomline of Jupiter’s Table to 4 15 14 1. When we search for his intention we see that the new bottomline is Latin gematria for D 1514 A. This mirrors the Durer monogram and date at the bottom ofthe engraving. The double occurrence of the date 1514 in Durer’s Table and in Durer’s monogramis therefore not a wild coincidence but a deliberate contrivance by Durer. It tells us somethng ofhis turn of mind that he could notice this combinatorial possibility. I suppose he systematicallycarried out all the permutations that preserve the gnomon property and scanned them for meaning.By whatever path, Durer overrode Agrippa’s Hebrew gematria with his own Latin gematria to puthimself into the magic square. This alerts us to look for other gematria and other differences withAgrippa.

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We need not look far. The sum of the whole table is 136. Pursuing the autobiographical hypothesis,I computed the Latin gematria for “Albrecht Durer,” ignoring the non-Latin umlaut. The sum is135. Since there is no “u” in the Latin alphabet, the name should really be “Albrecht Dvrer, ”but this would not change the sum, since “v” would then replace “u” as letter 21 of the alphabet.There is a significant discrepancy of 1 between 135 and 136. One must separate the 1 from therest of the table to make the sum “Albrecht Durer.” This amounts to a prediction: that Durer didso. Returning to the engraving to check this prediction, we see that he made the 1 unmistakablytaller than all the other numerals, as I did in transcribing the Durer Table above. This againdifferentiates the Durer Table from the Jupiter Table. In addition, one wing of the angel brushesthe 1 in the table, and only that numeral, verifying it divinity.

By splitting the sum into 136= 1 +135 Durer again puts himself into his own Table, next toGod. The magic square provides two more Durer signatures within a symbol of the divinity ofmathematics.

6 THE BAT OUT OF HELL

There is flying creature with the head of a mouse, the wings of a bat, and the tail of a lizard. Whodoes this hell-bat represent?

The perspective viewpoint of the engraving is on the horizon directly under his eye. He carries theeponymous motto “MELENCOLIA I.” The motto may well refer to a theory of melancholia of theFlorentine Neo-Platonist Marsilio Ficino (1433-1499) that Agrippa modified and promulgated [12].

Ficino and others of the time divided the world under God into three concentric spheres, roughlyTerrestrial, Celestial, and Intellectual. They also divided the human psyche into three faculties spe-cially suited to the corresponding spheres: mens imaginatio, mens ratio, and mens contemplatrix .Mens imaginatio empowered the artisan and the artist to view arrangements of the four Elementsin Space and Time, but nothing more. Since these were not believed to be the ultimate reality, webest capture the intention if we translate mens imaginatio rather literally, as Imagination, thoughtoday we are apt to call the process perception. Mens contemplatrix is the faculty that informsphilosophers of Plato’s Forms and theologians of angels, the presumptive occupants of the Intellec-tual Sphere. Let us render this faculty as Contemplation. Finally, mens ratio us the faculty thatenables the astronomer to predict the planetary motions. This is clearly Reason.

Ficino also kept the ancient psychology of the four humors, blood, choler, yellow bile, and blackbile or melancholia, one for each Element. Black bile seems to be a Greek myth invented to fit themental state of severe depression into the theory of the humors. Some said that the appendix andthe spleen secreted it; these structures secrete nothing. Astrologers had long associated Saturn,the slowest and darkest planet, with the least favorable humor, the worst mental state, the lowestElement, and therefore with the lowest people, those who work with or in the earth. Agrippaelevated Saturn, attributing to him the power to inspire all three faculties to the peak of creativity,and added geometers — “earth measurers”- – to the Saturnians.

Panofsky and Yates saw many Saturnian melancholic elements in the engraving, taking its melan-

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cholia in either the depressive or creative sense. They seem to think that Durer accepted Agrippa’sphilosophy, at least for the purpose of this engraving.

This is unlikely. In 1514 Copernicus (1473-1543) had not yet published his heliocentric cosmology,but the cardinal, mathematician, and philosopher Nicolas of Cusa (1401-1464) had quite demolishedthe Ptolemaic spheres. Durer’s role model, Leonardo, had written that the earth moves, not thesun, and that “those who have chosen to worship men as gods -as Jove, Saturn, Mars and thelike- have fallen into the gravest error,” had derided necromancers, and had excluded every trace ofastrology from his intensely rational writings on astronomy. Durer himself gave an ass’s ears and tailto the astrologer in his earlier woodcut, The Folly of Astrology , an illustration for the Narrenschyff(Ship of Fools, Sebastian Brant, 1494), and he surrounds his astrologer with a motley flock ofmisshapen fowl flying in random directions. I do not know exactly what these birds represent —the pronouncements of the astrologer? — but it is not complimentary. The bat-dragon too isinauspicious. It has the bat-wings and snake-tail of minions of the devil in works of Durer, such asthe dragon of St. George, so it is an emissary of the Prince of Lies, but it has the head of a mouse, soit is just a minor verminous nuisance, and it is fleeing the light. And by 1516 Durer had become anardent follower of Martin Luther, who linked Agrippa’s Occult Philosophy with witchcraft. Durerand Agrippa were on opposing sides in the three-way contest of the Neo-Platonic, Scholastic, andHumanistic philosophies.

The problem vanishes if the motto is ironic, expressing its own negation, a deeper and more matureexpression of the contempt for astrology already conveyed in Durer’s earlier woodcut caricature.Then Durer indeed referred to Agrippa, as Panofsky and Yates recognized, but only to ridiculehim. By giving the illiterate message “Melencolia I, ” to the hell-bat, Durer simultaneously beliesthe message and identifies the messenger as that itinerant pipsqueak Agrippa von Nettesheim whohad been nattering about melancholia in Nuremberg only recently.

Nevertheless the cosmology that Durer seems to use as his artistic language is related to that ofFicino. The tri-spherical Neo-Platonic cosmology and its medieval elaborations seem to have beenrather naive materialistic responses to Plato’s high poetry. Plato’s purpose in his cave allegorywas surely not to plant Forms in a sphere above the stars, but to say that there is a deeper realitybeyond appearances. Part of the Renaissance was exactly the rebirth of the original less naivecosmology of the ancients, many of whom were quite sure that the Earth moved. Free of boththe geocentric crystal spheres and the astrological influences of Ficino and Agrippa, da Vinci andDurer could still use the tri-spherical cosmology and its angels as an artistic language, and use theirexperimental cosmology when they spoke science.

Durer may even have felt a special relation to the Celestial Sphere. His coat of arms shows thefamily Doors above the clouds, and thus in the Celestial Sphere. Perhaps he also noticed that theroot of “celestial” names his preferred tool, the buren.

7 THE ANAGRAM

The classical Greek for black bile, transliterated, is Melancholia, from the roots of melas, black, andcholi , bile. The Latin is Melancholia, the German Melancholie. In no language is it Melencolia.

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Why this misspelling?

Maybe Durer simply spelled it wrong. The spelling of the time was quite phonetic. But he hada great passion for precise expression, great ambition for scholarly status, and a best friend whowas a scholar of classical languages. To assume that he could not spell “Melancholia,” would be asabsurd as supposing that the man who founded projective geometry could not draw a perspectiveof a box, or that the man who produced the first printed star chart did not know what a cometlooked like. It is more likely that he wrote and drew precisely what he meant, to the letter and thestroke.

One possibility is that he meant to mock the bearer of the motto by making him out to be illiterate.But we should not dismiss the possibility of a coded message. Anagrams were much more prominentin the 16th century than today. They were not merely riddles. They were also a standard wayof protecting intellectual property. Thinkers of the time who solved an important problem forexample, a mathematician who discovered how to solve some equation, or a chemist who learnedhow to make an explosive did not stake their claim for posterity by publishing or patenting theresult. The system of scientific archives was over a century in the future. Some protected theirdiscoveries by scrambling its letters and publishing the jumble together with the problem. Thosewho knew the secret could read the jumble, and then they would see that the author had alreadyknown it. Roger Bacon had concealed the formula for gunpowder in this way in order to preventthe proliferation of this terrible weapon. Such a meaningless jumble of letters declared on its facethat it was a cipher. The next degree of concealment was an anagram, whose scrambled lettersform a different cover message. This can conceal not only the content but even the existence of ahidden message. In the third degree of concealment, the cover message is also plausible.

The misspelling of the motto could be the constraint imposed by such an anagram, where theletters are given and a word has to be made from them willy-nilly. If so, it has done its job well.Only the cover message has been read in recent centuries.

To unscramble it I proceeded as follows. Since so many Durers, father, mother, and repeated son,already hide in the engraving, I guessed that the motto might hide another. This amounts to aprediction. To test it I went to Durer’s coat of arms to see how he might depict himself. ThereI found if I did not invent the caelo rebus by which Durer represents his art. CAELO indeedfits into MELENCOLIA. The leftover letters quickly arrange themmselves into the common nounLIMEN, commonly meaning gateway, doorway, threshold, lintel, walls, house, home, boundarypath, and limit, according to context. MELENCOLIA then decodes to LIMEN CAELO, gatewayin heaven. This describes the Durer coat of arms itself quite accurately, fulfilling the predictionthat the anagram hid a name for Durer. It indirectly supports the rebus theory of the coat ofarms. It also applies well to the dim archway in the heavens that frames it, and will acquire furthermeaning as we go.

The speed with which this prediction checked out suggested that I read Durer correctly. Theproposition before us is that Durer constructed the motto MELENCOLIA I from the covert oneLIMEN CAELO I, put melancholic elements and the mooonbow to fit them, and added the hell-batto signal that the cover message was ironic.

“Limen caelo” is as positive in spirit as “Melencolia” is negative. The bright spirit of the covert

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message and the dark spirit of the picture. pull in opposite directions. This tension will grow aswe go further until it tears the picture open.

8 THE NUMBER I

What does the number I in the motto stand for? Some have suggested that it refers to the medievaltripartite universe, and expect that there were engravings of Melencolia II and III, perhaps lost.

Melencolia I is indeed one of three engravings of Durer, called his Masterpieces, that make aplausible triptych of the three realms of Imagination, Reason, and Contemplation. The other twoare The Knight, Death, and the Devil , and St. Jerome in His Study . We may test our understandingof Durer’s language further on his knight. He portrays the knight as spectacularly oblivious to Deathand the Devil, though in all other such encounters in the work of Durer, the knight engages evilin combat. It is one thing to ignore a bat out of hell, another to ignore both Death and the Devilthemselves.

The knight is generally believed to represent the ideal Christian Soldier described by Erasmus(1466?-1536), who was a revered acquaintance of Durer. One can be more specific. Durer wrote afervent plea to Erasmus, perhaps never sent, to join with him and Luther in battle against the “Caveof Hell,” and die a martyr’s death. Erasmus, no believer in abstract ethical principle, had remainedwithin the Catholic church and worked for reform instead of leaving or applying for martyrdom.Presumably the beautiful knight surrounded by Death and the Devil was read by viewers of theday as Erasmus himself, surrounded by clerical corruption and seemingly ignoring it.

I f so, Durer pays Erasmus a notable token of respect. Part of the garment of Death forms anagonized head smashed between the Knight’s two fists. This face is evidently part of the story.It says that the knight Erasmus only feigns blindness to the evil around him while he covertlyrecognizes and crushes it. This increases our confidence that the hidden faces sometimes furtherthe story.

Eventually the high-minded Durer would lament that the behavior of the Protestant clergy madeeven the Catholic clergy look respectable by comparison. In the end he kept his high esteem forErasmus. His 1526 portrait Erasmus of Rotterdam has a hidden face high on the left sleeve ofErasmus. It is gleeful.

The triptych theory, however, does not explain why one picture was captioned and not the othertwo, and so does not explain the I. I suggest another reading of the I.

For a mathematician like Durer, an “x primus” more often suggests an earlier x than a later xsecundus or x tertius. This amounts to a prediction, which I tested by going to Durer’s course onmeasurement and searching for letters followed by numerals. There is no occurrence of “a0” or a′

(for example) but there are five of a1. Every a1 is associated with a separate appearance of a, butonly one of them with an occurrence of a2, and none with a3. This rather supports our theory ofthe I and suggests a further prediction: that the limen caelo I , the new gateway in heaven, willbe found preceded in this very engraving by the original limen caelo.

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9 THE PUTTO

Who is the putto and what is he doing?

A second treatment of the suffix “I” fits so well that I suspect that Durer used both. If one stirsthis letter in with the other ten, the only anagram to emerge that contains the word “caelo” is“Caelo limine,” which can mean “I engrave at the wall or “at the edge.” And indeed the instrumentwielded by the putto has a cross-bar at its top. The artist presses against the cross-bar with hispalm to grave deeper lines. It is not a pen or a stylus, therefore, but indeed a graver, the celestialburin again. The putto indeed leans nonchalantly against a wall as and sits on the edge of themillstone as he engraves. This could be a multiple coincidence, but it seems more likely that thisis a second communication from Durer, announcing that the putto is yet another symbol for Durerin the engraving, as Yates said, not the generic starving artist scribbling on a slate, as Panofskyproposed. On the other hand it fits Panofsky’s conclusion that Durer meant this work to showmany aspects of himself. In the tripartite psychology the artist’s special faculty is Imagination. Ifthe putto is Durer’s Imagination, we should be alert for his Reason and Contemplation as well.

Yates suggested that what the putto is engraving is “Melencolia I” itself, in a self-referential way.But a close look at the sight lines in the engraving shows that the putto is looking over his tabletdirectly at the dog. Presumably the putto is engraving the dog on his tablet. Durer’s engravingsof animals are among his finest works.

10 THE ANGEL

Who is the angel?

Evidently Durer uses this angel’s wings not to indicate celebrity like the eagle-wings in his familycrest, but to sanctify. Her wings connect the holy numeral 1 in Durer’s Table, the hour-glass, thescales, and nothing else, blessing the scientific instruments above the artisan tools that litter theground. She holds a sealed book and a pair of compasses. The angel may well be the facultyof Contemplation, perhaps Durer’s own. She might also be Geometria, the spirit of geometry,for whenever the quadrivia are personified, Geometria is the one with the compasses. But TheAstronomer , a Durer woodcut of 1504, shows an astronomer measuring a globe with compassesunder a full moon, so this interpretation is insecure.

Some say the angel has measured the stone globe that lies before the dog. But the astronomerstudies his globe intently while she ignores the globe, looks right past it — check the line of sight— as she should if she is Contemplation and the globe is the Terrestrial Sphere, the realm ofImagination.

Most commentators also say that the angel is melancholy, or is even the spirit Melancholia herself.They are probably influenced by the motto on the banderole, the shadow on the angel’s face, andthe fist on her cheek, traditional signs of melancholia. But the deciphered motto, we have seen,is not negative but positive and in any case is not over her but over the Solid. Her expression is

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alert, focused, and subtly smiling, not soft and sad. Her gaze is not downcast, as was mandatoryfor portraits of melancholia, but slightly elevated. Her pose indicates a Thinker more than aMelancholic.

If she had melancholia, it would have to be the creative form of the humor, not the depressive. Infact she is visibly not creating. Her compasses are held in a way that puts them out of service.Her book is sealed. The unresolved tension between her positive expression and attitude and theapparently negative legend is part of the hold that the engraving has on us.

She looks at nothing in the scene. The polyhedron, the putto, and the dog are directly to her right,and the globe is beneath her line of sight. She looks up out of the frame, right past what is going onin the sky behind her, meteor, hell-bat, moonbow and all. This is consistent with her representingthe faculty of Contemplation that connects its user to the Intellectual Sphere of Forms and angels.

What the angel is doing is remarkable. She is doing nothing.

I propose that her main function in this engraving is that of the knight between Death and theDevil: to ignore evil. She sees nothing and does nothing. She is unlike the putto, who studies thedog intentlly and draws it. But neither see the bizarre night sky.

11 THE LADDER

Where does the ladder lead? This is easy. Any artist who put a topless ladder next to an angel, ahouse, and an up-turned stone in 1514 was sure that the educated viewer of the time would see areference to Jacob’s ladder, angel, house, and stone in Genesis 28:12, which concludes,

And he [Jacob] was afraid, and said, How dreadful is this place! this is none other butthe house of God, and this is the gate of heaven.

The house shows no door, so the gate of heaven must be the ladder for Durer as for the author ofGenesis. Ladders as gates of heaven occurred commonly in earlier art, including Durer’s own. Wehave found the original limen caelo. This makes yet another hidden gate for Mr. Gates. Thanksto the coat of arms and its porta caelo, the gate of heaven, porta caeli , suggests the house of Dureras well as the house of God.

12 THE SCALES

What are the scales for?

The balance is one of the few scientific instruments in the picture. They all hang on the wall and theartisan tools litter the yard, as the experimental philosopher is sanctified above the craftsman. The

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scales hang on the side wall between the angel and the putto, level and balanced. One dish touchesthe putto, the other the angel. There is a balance between putto and angel, first literally, thereit hangs between them, and then metaphorically, they have equal weight in some sense; perhapsequal divinity. The putto-angel equation seems to be a literally central message of the engraving.This fully supports Yate’s interpretation of the triumphant artist and of a balance between theIntellectual and Terrestrial spheres represented by the angel and the putto.

Albrecht confirms this central equation by setting the eyes of the putto level with the eyes of theangel. If the main message were the inferiority of the artist, Albrecht would have set the puttobelow the angel. As though to emphasize that this is not the case, the angel is actually lower onthe paper than the putto, but this is just the unusual perspective. We are looking slightly down atthe two and the angel is nearer to us than the putto. By following lines of the house to the horizonwe see that the center of attention, the vanishing point, is unusually far left. It is the point on thehorizon beneath the eye of the hell-bat. The line joining the eyes of the putto and the angel goesthrough the vanishing point. If this represents a line parallel to the side of the house, then it toois horizontal. The angel and the putto, Contemplation and Imagination, attain equal heights andtherefore equal sanctity.

Nevertheless Durer is not even-handed between putto and angel. The angel is seated on thefloor of the yard, the putto on an elevated millstone. While he works, she idles, elaborately andconspicuously. Her book is sealed, her compasses are so held that they cannot be in use, and if sheis looking at anything, it is not in this engraving. As in The Knight, Death, and the Devil , doingnothing may signify a failing in the eyes of Durer.

13 THE HOUSE

Whose house is this?

The house too is paradoxical and meaningfully polymorphic. From up close we see two blank wallssupporting a hodge-podge of instruments. The side wall does not seem much wider than the puttoleaning against it. It is about the right thickness for an out-house or a chimney, not a residence.Its front wall extends indefinitely to the right and holds a bell, Durer’s Table, an hour-glass, and asun-dial. Its left edge bisects the picture to the millimeter.

Step back three or four paces, however, or squint, and the scale changes. The magic square andbell mutate into a lattice window, the hour-glass into a bay window, the scales into a side window,and the out-house into a full-sized house showing us three of its windows.

Assembling familiar objects from unrelated parts in this way, for example, faces from fruits, wasan optical illusion often practiced by artists of the era, but here there is allusion as well as illusion.If the house is what the sanctified instruments on its walls open into, it is the humanly perceptibleuniverse, the entire physical cosmos. The ladder and the angel have already told us that this isalso the house of God, and the instrumental windows tell us that measurement and mathematics,not the scriptures and the classics, are our way of seeing into it. This is not the Agrippan heresyat all but an opposite one, the battle flag of Renaissance science.

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Then there is the question of the doors to the house. The problem is that there are none. Thehouse that we see as we step back has windows of mathematics, time, and weight, but no door.When the Man-of-the-Door leaves the doors off the House of God, he is trying to say something.

If the house indeed represents the cosmos, the absence of doors probably symbolizes that we haveno direct access to absolute reality. This form of relativism was a common idea by Durer’s time. Itwas part of the doctrine of learned ignorance of Nicholas of Cusa, and is implicit in Erasmus andhis praise of folly. Durer could expect his viewers to be able to make out this message.

Before he did this engraving he had already written that the human mind cannot know absolutebeauty, and in this picture he may have said the same about absolute truth. Indeed, we must notassume that he separated truth and beauty as cleanly as some people claim to do today, especiallysince he expected a mathematical theory of both. For example, the Latin word scientia translatesbest as “art” and the Latin ars is often translated as “science,” as though the two words havewaltzed about each other down the centuries. From the Terrestrial Sphere we can look at realityPerhaps it can be entered by the ladder, but that is traditionally reserved for angels.

14 THE MILLSTONE

What did the millstone grind?

The geometric center of the picture falls just under the putto and within the tilted stone wheel.Panofsky and Yates call this a grindstone, and Doorly calls it a millstone. Its perimeter is brokenwhile its face is smooth. It would make a passable millstone but a miserable tooth-rattling grind-stone, so presumably Doorly is right. It is the only stone in the engraving that Jacob could useboth as pillow and pillar. By elimination, however, we have inferred that it might also representthe Celestial Sphere, the realm of Reason. We have already identified Durer’s Imagination as theputto, and his Contemplation as the angel, so the small silhouette in the center of the millstonemight be his Reason. Its size need not be interpreted as a sudden attack of modesty on Durer’spart, if it puts him at the very center of the Celestial Sphere.

If the millstone represents the Celestial Sphere, the seating of putto and angel probably has meaning.The angel sits on the stone floor of the workyard, practically on the level of the Terrestrial Sphere.The putto perches naturally on the rim of the Celestial Sphere as he draws busily. The practicalexperimenter, with the artisan’s faculty of Imagination, experiments with matter in space andtime, but not the philosopher/theologian, endowed with the faculty of Contemplation. Neverthelessaccording to the experimental philosophy these experiments are the source of knowledge, and sothis engraving puts him above the Celestial Sphere and therefore into the Intellectual Sphere, theabode of angels and Forms, while it grounds the faculty of Reason. Durer puts the experimentalphilosophy into his seating arrangements.

This raises the question of why Durer would represent the celestial sphere by a millstone. Accordingto some scholars, much older Icelandic sagas already represented the Celestial Sphere by a millstone,one that flew off its axle to account for the precession of the equinoxes [13]. If Durer’s millstoneis Hamlet’s, then it grinds salt for the sea. Indeed, Melencolia I and Hamlet have several striking

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elements in common: the parental nocturnal ghosts, the melancholy in the works themselves, andthe fact that both were created in a year of mourning, Durer’s for his mother and Shakespeare’sfor his son Hamnet.

15 THE SOLID

How and why was the Solid constructed?

Durer seems to put his Solid in a place of honor. The artist stands before it, the ghosts occupy it,the motto names it. Yet the Solid seems bleak and vapid against the angel, and its shape is subtlyambiguous. One looks for a deeper significance to enrich it and justify it artistically, or at least todecide its shape.

A rhombus is a planar quadrilateral whose sides are all equal. A rhombohedron is a closed surfaceformed of six congruent rhombuses. Most people see the Solid as a truncated rhombohedron withits longer axis vertical. If the two missing vertices are restored, they lie in a vertical line on thepaper, and the compulsion to see the figure as a rhombohedron is irresistible.

In principle, however, one cannot determine the form of a solid from one perspective view. Some-times the cues to the interpretation are so strong and unconscious that we cannot help inferring aparticular solid. For example, it is hard not to see a perspective of a cube as a cube, or a perspec-tive of a person as a person. Nevertheless one can shift any surface point along the line of sightfrom the eye without changing its apparent location in the perspective view. Therefore there aremany solids all having the same perspective outline, and some have much the same shading as well.The Reversing Staircase Illusion exploits this ambiguity. So does the Drer Solid. There are twodominant views.

If I view the Solid with my head erect like the hidden face of the woman, I have to see it as atruncated rhombohedron.

If I view the Solid with head cocked so as to see the hidden face of the man, then I can see the Solidas a nearly rectangular slab with two diagonally opposite corners trimmed, cocked and cantileveredtoward the horizon. I owe this interpretation to Dr. Basimah Khulusi [9].

Some speak of measuring the angles of the Solid from the engraving. These angles depend onsecondary assumptions that are not forced on us by the engraving. The Solid is a puzzle that is inprinciple unsolvable without another view. I suspect that is why the angel is smiling.

One valid interpretation presented by MathWorld on the internet sees the acute angle of the rhom-bus to be a fifth of a circle [20]. That would bring the Divine Proportion ϕ = 1.618 . . . into theengraving. If one compresses the engraving vertically by a factor of

√ϕ, its outline becomes square,

and the Solid becomes an unmistakable truncated cube. I suspect Durer deliberately made thatinterpretation possible.

Another apparently valid interpretation sees the same angle as close to 80◦ [11]. This results from

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measurements on the engraving assuming that certain lengths are show in true proportion. Butthere is no way to tell from the engraving if a given length is shown in true proportion. Thisrequires an additional assumption.

The question of why Durer chose to truncate a rhombohedron instead of another solid we take upsoon.

Before the engraving he apparently made a rough sketch of the Solid that has survived, though ithas been recognized as that only recently [16]. It shows an irregular pentahedron inscribed in acircle. The apex angle in the drawing is indeed 79.5◦± .5◦. The agreement with one interpretationof the engraving seems noteworthy.. The question remains: How did Durer arrive at that angle?

I note that the circle in the drawing put forward by Weitzel has been divided by a compass intoseven equal arcs as if to inscribe a regular heptagon. There is no Euclid construction for this,but Durer gives an excellent approximation to the regular heptagon in his course on measurement,without telling us that it is approximate.

The vertices of the irregular pentagon seem to be five of the seven vertices of the heptagon. Thetwo long edges of the pentagon are chords of the heptagon that connect the apical vertex to itsnext-nearest neighbors on each side, The remaining three edges of the pentagon are edges of theheptagon joining the remaining vertices. This would lead to an apical angle of 77.2◦ theoretically.

If the Solid, like the millstone, represents a Sphere of the Neo-Platonic cosmology, then the mathe-matical nature of the Solid, and its height in the picture, suggest that it represents the IntellectualSphere, the third realm. This is the abode of Platonic Ideas and angels. The ghosts in the Solidalso support this conclusion, for this puts them in a philosopher’s version of Heaven.

Leonardo had already undertaken a mathematical theory of beauty. That goal would occupymuch of Durer’s later years, and result in works on human proportions and on constructions ofprojections and perspectives with compass and ruler. Durer’s study of human proportions is anamazingly dry gallery of stark outline drawings of standing human nudes and tables of anatomicaldimensions to three decimal places. It may never have been of much use to artists, but it fulfilledother functions for him. Human measurements were to be the foundation of a geometry of beauty,as stellar measurements were the foundations of Ptolemy’s geometry of the heavens. Music wasalready divine and part of the quadrivium because it was considered mathematical. By providinga mathematical basis for art, Durer hoped to sanctify the graphic arts as well. Possibly thisbelief in a mathematical art was only one aspect of a belief in a general mathematical wisdom, amathesis. Then the Solid does not simply show off Durer’s mathematical muscle, it represents themathematical nature of the Intellectual Sphere, the divinely inspired mathesis that many sought,and its inaccessibility to human cognition. We will see a deeper meaning later.

16 THE HEXAGRAM

There is a hexagram or Shield of David built into the Solid. Has it any significance?

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Durer’s Neo-Platonic Hebraicism is conspicuous in his work. He used gematria in this picture. Herevered St. Jerome. In his several drawings of Jerome studying Latin, Greek, and Hebrew Bibles,he placed the Hebrew Bible conspicuously above the others, and higher is holier. There is a furtherHebraicism just below the surface in the Durer Solid.

Recall that Durer was a descriptive geometer, skilled in constructing views of bodies from all sides.We should look at the Solid from all sides while we are searching for meanings. Viewed from the skyas a truncated rhombohedron, the edges of the Solid form a Shield of David framed in a hexagon.The top and bottom triangles of the Solid project into the two crossed triangles of the hexagram.Perhaps Durer truncated the cube, rather than some other regular solid, in order to create thishexagram.

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It would be anachronistic, however, to call this hexagram a Jewish star or to infer some NorthGerman philo-Judaism. To be sure, centuries earlier the Khazars of the Crimea had adopted thehexagram for their flag when they adopted Judaism, but in 1514 Nuremberg the hexagram wasstill mainly a magical device, an amulet, possibly referring to the Hebrews of the Bible but not tocontemporary Jews, and more often found in churches than synagogues. The Shield of David wascalled that because it was supposed to shield its bearer from evil spirits. Some called the hexagramthe Seal of Solomon, and others applied these terms to the pentagram. Traditions firmed later thatgave Solomon’s star five points to David’s six, and stood Solomon’s on two points to David’s one.A two-footed hexagram like Durer’s ground plan is seen on a German altar of Durer’s time. Notlong after Durer’s death, the Jewish community of Prague was granted the privilege of a flag, andchose the hexagram, furthering its evolution from magic charm to Judaic symbol. Since Solomonwas the famous temple builder, not David, Durer might have thought of his hexagram as a Seal ofSolomon.

In short, Durer may have circumcised the cube to Hebraicize it, to consecrate his new IntellectualSphere.

17 THE BALL

What is the stone ball?

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In Durer’s representations of the goddesses Fortuna and Nemesis, both stand on globes, liable toroll in any direction at the least provocation, to convey the unpredictability of Luck and Fate.The reign of chance distinguishes the Terrestrial Sphere from the Celestial, which is governed bymathematical law. The stone ball seems to represent the Terrestrial Sphere. It is indeed thelowest of the three stones in this picture as the Durer Solid is the highest, confirming that theyrepresent Durer’s version of the tripartite cosmology. The three stone forms also verify that Durer,like Nicolas of Cusa and Leonardo, had replaced the naive tri-spherical geocentric Neo-Platonistcosmology by a tri-partite philosophy with three modes of knowledge, no longer geocentric.

Durer seems to concern himself with the relative status of angel and putto, Contemplation andImagination. He balances them with the scales, but separates them in status by their seating andtheir activity. The angel practically sits on the Earth, at the level of the Terrestrial Sphere. Theputto, however, perches on the highest point of what we have read as the Celestial Sphere. If wehad not already seen Durer deify himself, we would be surer that we were reading this wrongly.

18 DURER SPACE AND TIME

What does Durer express with his space and how does he express time?

He uses the vertical dimension in a traditional way. Higher is holier. Jacob’s angels get to heavenwith a ladder, not a shovel. Angel and putto have equal divinity because they have equal altitude.The tools of artisans litter the ground, so they are profane, while the instruments of science andmathematics hang carefully on the wall nearer to the sky, so they are more sacred.

It seems that Drer’s horizontal axis represents an aspect of time. On the right is the old limencaelo, the ancient Biblical gate of heaven, and on the left is the new mathetic gate of heaven. Thenthe left edge of the house, which neatly bisects the picture, also separates the old order from thenew. The near eye of the putto lies just on this center line, making it clear who the central figureis.

According to this theory, the triptych should have Jerome on the right, Erasmus in the center,and Melencolai I — Durer in all his forms — on the left. The vanishing points of these threeMasterworks also form a plausible sequence: far right, center, and far left respectively, as if theartist progressed from the past to the future.

Thanks to this new-fangled Italian refinement of perspective, there is also a third dimension inthe picture, that of depth into the scene, distance along the line of sight. Durer seems to use thisexpressively too. On the right side our view is blocked and on the left there is an infinite vista. Itseems that perspective depth in the picture is the familiar metaphor for depth of vision.

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19 THE COMET

What is the comet doing there?

It rather resembles some recent comet photographs. Apparently this one blots out the stars.

The great comet 1471Y1 was first seen on Christmas Day in Durer’s birth year, and Durer wroteof seeing a comet in 1503. The physical natures of meteors and comets were not yet known in1514. Even Galileo would still believe that comets were formed from atmospheric vapors leavingthe Earth. But da Vinci already mocked the idea that events in the sky foretold events on Earth,and so did Durer.

Since the Bible has already explained many elements of the picture, let us ask it for this too. Theanswer is immediate and unique. The only blinding light in the biblical sky is the original light of“Fiat lux,” representing divine creation. Then the sky recalls the time in the Biblical story afterthe creation of light but before the creation of the stars.

We have already read the moonbow as God’s memo to himself to send no more genocidal floods.The light and the bow in the clouds of Genesis represent two of the greatest gifts of God. Thepicture should be joyous but it is melancholy. The tension grows.

20 THE NIGHT

Why is it night?

We have arrived at a paradox that can be productive. The left side of the picture represents thecreative divine spirit, yet is set in the gloom of night. Durer has even transformed the rainbow ofthe common understanding into a moonbow. The bright light in the sky illuminates nothing; themoon behind us casts all the shadows and creates the moonbow.

Initially I took the crucible to the left of the Solid to represent the science of alchemy or chemistry,which were not yet clearly differentiated. Doorly, however, identified it as a goldsmith’s crucible.If it were an instrument of science it would be elevated, but it is on the stone floor, though theperspective puts it high on the paper. So Doorly is right. The crucible may be an eternal flamebetokening Durer’s love for his father the goldsmith, but it is not an instrument of science.

But this means that there is no experimental science in the left side of the picture, only crafts. Allthe experimental instruments are on the right. This is consistent with the interpretation of the leftside as a representation of direct access to the thought of the Creator, who is evidently a Geometer,through mathematics. This would then be a precedent for the famous inspiration of Kepler, thatthe work of the scientist was to rethink the creative thought of God.

This intoxicating idea is still active today. Having put it on the table, I am obliged to attach a warn-ing label: “Do not take internally. May result in bad physics.” For example, Kepler experienced

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this enthusaism when he related the Platonic solids to the planetary orbits. quite erroneously, butnot when he discovered his three planetary laws, presumably because they did not fit his conceptionof divine geometric law. Principles that scientists attribute to God often resemble suspiciously theearliest principles they ever learned, absorbed when they were least critical and most vulnerable.They are generally projections of the kind that we are trying not to make in our reading of thispicture. Experiment and observation, as in Leonardo’s dictum that “All knowledge comes fromexperiment” and Durer’s detailed depictions of nature, are shields against such Promethean hubris.

While Leonardo and Durer were both mathematical optimists, they differed importantly in theiroptimism. Leonardo wrote confidently of “a complete knowledge of all the parts, which, whencombined, compose the totality of the thing which ought to be loved.” Durer on the contrary saidshortly before this engraving, “But what absolute beauty is I know not. Nobody knows it but God.”In the left half of this picture Durer represents such a complete geometric knowledge. It lies onthe same long line of distinguished fantasies of mathematical wisdom as the number mysticism ofPythagoras and Plato, the Ars Magna of Ramon Llull and Giordano Bruno, the vision of MathesisUniversalis that Descartes and Leibniz shared, and the Ars Combinatorix of Leibniz. Durer mayhave shared with Llull before and Leibniz later the recurrent dream of people sitting down togetherto calculate the truth about God instead of slaughtering each other for it. If limen caelo on the rightis the old theocratic order, then limen caelo I on the left present a new more secular mathematicalorder.

But Durer’s pessimism takes him off this line. Years after, Durer renounced even the goal ofabsolute truth as futile and hopeless, crying out,

“The lie is in our understanding, and darkness is so firmly entrenched in our mind thateven our groping will fail” [Panofsky 1971].

The engraving should indeed be read as an illustration as Doorly says, and it illustrates Durer’sown lamentation. It contrasts the absolute truth and our ignorance of the absolute truth. No onein the engraving sees the light, no one seems to know that it is there. The scene is set at night tocapture our benighted ignorance. The darkness in the engraving is the darkness “firmly intrenchedin our mind.” Even the impenetrable mystery of the Solid is a good metaphor for our inability tosee the Divine Truth from our limited perspective.

But no one in the engraving looks melancholy, except possibly the dog. They are not even awareof their ignorance. The melancholy one is the engraver himself, who sees the darkness firmlyentrenched in our mind. In this pessimistic estimate of the limits of human understanding, perhapsinfluenced by Pico and Erasmus, Durer took a step toward modernity, from hubris to humility, thatthe more optimistic Leonardo did not. Much as Panofsky said, the melancholy is Durer’s and isdue to his perception of the limits of human understanding, expressed by the missing doors intothe house of God, the ignored Light, the ambiguity of the Solid.

His emotional growth did not stop here, however. His posthumous publication on perspectiveshows that he did not abandon the search for a geometry of beauty but brought it down to earth.It includes a woodcut that is again divided by a vertical line into two dramatically different halves,with flesh on one side and knowledge on the other, as though referring back to Melencolia I , but

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now the picture is simpler, and shows no melancholy, only diligence and acceptance and humor.The flesh is represented by a voluptuous reclining woman, not an angel. The vertical division is nowthe screen of Durer’s perspective machine, not the edge of the house of God. And now knowledgeis represented by a lean artist, now demoted to draughtsman, industriously drawing the woman inaccurate perspective. The eyes of the artist are still slightly higher than those of his subject, buthe sits on a chair and she is on a distinctly higher level, an astonishing token of humility for Durer.

Durer has not lost his sense of humor. We have a standard frontal view of the odalisque but thedraughtsman limits his perspective view to her bottom end. Human knowledge is limited indeed.

Gone is the mystical mathetic vision of absolute truth, the perspective of infinity, but also theself-deification and the gloom. Both halves of the picture live on Earth. Knowledge, with all itslimits, is no longer a light in the night sky but a record of actual experience.

21 ACKNOWLEDGMENTS

These amateurish thoughts on Melencolia I would have been impossible without the scholarship ofErwin Panofsky and Francis Yates. I have reveled in their studies of Durer and in Yates studies ofthe Hermetic tradition. I have also drawn on the study of Hamlet’s mill by Giorgio de Santillana andHertha von Dechend, and on a study of Hamlet by Stephen Greenblatt. The heraldry dictionaryof Cecil Wade helped with Durer’s coat of arms. I benefited from the gematria on the website ofthe Aiwaz Gallery, though its arithmetic was not always perfect. My thanks go to Roy Skodnickfor introducing me to the work of Yates, to Prof. Heinrich Saller for the print of Melencolia Ithat triggered this process, to Rabbi Mario Karpuj for reminding me that Jacob’s ladder is thegate of heaven, to Shalom Goldman for stimulating, informative, and encouraging discussions,the reference to Yates on Durer, and the meaning of the Magen David in the Renaissance; toDanny Lunsford for refining the Schreiber construction of Durer’s Solid and for the reference tohttp://meteoros.de/halo/halo1.htm; to Dr. Basimah Khulusi for helpful discussions and a model ofthe Durer Solid and pointing out the asymmetric interpretation, to Carla Singer for her expertisein art history; and to Shlomit Ritz Finkelstein, for lively discussions, help with Torah, and usefulsuggestions. An earlier stage of this study was published in The St. Ann’s Review .

References

[1] Heinrich Cornelius Agrippa: Of Occult Philosophy, digital edition, ed. Joseph H. Peterson,http://www.esotericarchives.com/agrippa/agrippa1.html. Based on the translation of ”J.F.”,printed by Moule, London (1651).

[2] Doorly, Patrick. Durer’s Melencolia I : Plato’s Abandoned Search for the Beautiful. The ArtBulletin88, 255 (2004)

[3] Durer, Albrecht. Vnderweysung der Messung mit dem Zirckel un Richtscheyt in Linien, Ebnenund Gantzen Corporen, Vol. 1-4. Nuremberg 1525. Republished with English translation of

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Walter L. Strauss, revised, on compact disk. Editor, anonymous. Commentator, David Price.Octavo Editions, Oakland CA, 2003.

[4] Finkelstein, David Ritz. Melencolia I: Durer Decoded, The St. Ann’s Review , 2004 (in press).

[5] Galileo Galilei, Dialogue on the Great World Systems. The Salisbury translation. Revised, ,annnotated, and with an Introduction by G. de Santillana. University of Chicago Press (1953)

[6] Goldman, Shalom. The Sacred Tongue; also http://uncpress.unc.edu/booksT-6957.html. Forthe concept of Hebraicism.

[7] Greenblatt, Stephen. Will in the World: How Shakespeare Became Shakespeare. Norton(2004). On Hamnet and Hamlet.

[8] Heckscher, William S. Melancholia (1541). An Essay in the Rhetoric of Description by JoachimCamerarius. In: Frank Baron, ed., Joachim Camerarius (1500-1574). Beitrge zur Geschichtedes Humanismus im Zeitalter der Reformation. Essays on the History of Humanism Duringthe Reformation. Humanistische Bibliothek, Abhandlungen 24 (Munich, 1978): 32-103

[9] Khulusi, Basimah. Personal communication.

[10] Lunsford, Daniel. Personal communications concerning the Durer Solid and Ratio (2004).

[11] MacGillavry, C. The polyhedron in A. Durer’s engraving Melencolia I. Nederl. Akad. Weten-schap. Proc. Series B 84, 287-232 (1981).

[12] Panofsky, Erwin. The Life and Art of Albrecht Durer . Princeton University Press, 1971. Thestandard work. It reproduces the Durer works mentioned here.

[13] Santillana, Giorgio de, and Hertha von Dechend. Hamlet’s Mill: An Essay on Myth and theFrame of Time. Gambit, 1969. Argumentation for the astronomical interpretation of all myth.

[14] Peter Schreiber. “A New Hypothesis on Durer’s Enigmatic Polyhedron in His Copper En-graving Melancholia I.” Historia Mathematica 26, 369-377, 1999.

[15] W. Cecil Wade. The Symbolisms of Heraldry or A Treatise on the Meanings and Derivationsof Armorial Bearings. London: George Redway, 1898.

[16] Weitzel. H. A further hypothesis on the polyhedron of A. Durer’s engraving Melencolia I.Historia Mathematica 31 (2004)

[17] Frances A. Yates. The Occult Philosophy in the Elizabethan Age. London ; Boston : Routledge& K. Paul, 1979. Especially chapter 6.

[18] Frances A. Yates. The Occult Philosophy in the Elizabethan Age. London ; Boston : Routledge& K. Paul, 1979. Especially chapter 6.

[19] http://gallery.euroweb.hu/html/d/durer/ shows all the Durer works I mention.

[20] http://mathworld.wolfram.com/DuerersSolid.html describes and names the Durer Solid.

[21] http://meteoros.de/halo/halo1.htm is a remarkable gallery of portrayals of heavenly phenom-ena like moonbows and moon halos, including several from Durer’s time.

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